Error Models Lecture 06 - MIT by pptfiles


									Error Models
 Lecture 06
Thomas Herring
          Issues in GPS Error Analysis
• What are the sources of the errors ?
• How much of the error can we remove by
  better modeling ?
• Do we have enough information to infer the
  uncertainties from the data ?
• What mathematical tools can we use to
  represent the errors and uncertainties ?

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  Determining the Uncertainties of GPS
         Parameter Estimates
• Rigorous estimate of uncertainties requires full
  knowledge of the error spectrum—both temporal
  and spatial correlations (never possible)
• Sufficient approximations are often available by
  examining time series (phase and/or position)
  and reweighting data
• Whatever the assumed error model and tools
  used to implement it, external validation is
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                     Sources of Error
• Signal propagation effects
      –   Receiver noise
      –   Ionospheric effects
      –   Signal scattering ( antenna phase center / multipath )
      –   Atmospheric delay (mainly water vapor)
• Unmodeled motions of the station
      – Monument instability
      – Loading of the crust by atmosphere, oceans, and
        surface water
• Unmodeled motions of the satellites
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 Simple geometry for incidence
 of a direct and reflected signal

      Multipath contributions to observed phase for an antenna at heights (a)
      0.15 m, (b) 0.6 m, and (c ) 1 m. [From Elosegui et al, 1995]

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                  Characterizing Phase Noise


                                                           0 mm

          1       2        3        4            5 Hours
          Elevation angle and phase residuals for single satellite
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                                 the Noise in
                                 Daily Position

                                 Note temporal
                                 correlations of 30-
                                 100 days and
                                 seasonal terms

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                 Spectral Analysis of the Time Series to
                 Estimate an Error Model

              Figure 5 from Williams et al [2004]:
              Power spectrum for common-mode
              error in the SOPAC regional SCIGN
              analysis. Lines are best-fit WN + FN
              models (solid=mean ampl;

              Note lack of taper and misfit for
              periods > 1 yr
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          . . . spectral analysis approach
• Power law: slope of line fit to spectrum
      – 0 = white noise
      – -1 = flicker noise
      – -2 = random walk
• Non-integer spectral index (e.g. “fraction white
  noise”  1 > k > -1 )
• Good discussion in Williams [2003]
• Problems:
      – Computationally intensive
      – No model captures reliably the lowest-frequency part
        of the spectrum
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Examples of times series and spectra for global stations
From Mao et al., 1999
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          Short-cut: Use white noise statistics ( wrms) to predict the flicker noise

               White noise vs flicker noise from Mao et al. [1999] spectral
               analysis of 23 global stations
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          “Realistic Sigma” Algorithm for
              Velocity Uncertainties
• Motivation: computational efficiency, handle time series with
  varying lengths and data gaps; obtain a model that can be used in
• Concept: The departure from a white-noise (sqrt n) reduction in
  noise with averaging provides a measure of correlated noise.
• Implementation:
      – Fit the values of chi2 vs averaging time to the exponential function
        expected for a first-order Gauss-Markov (FOGM) process (amplitude,
        correlation time)
      – Use the chi2 value for infinite averaging time predicted from this
        model to scale the white-noise sigma estimates from the original fit
      –      and/or
      – Fit the values to a FOGM with infinite averaging time (i.e., random
        walk) and use these estimates as input to globk (mar_neu command)

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                 Effect of Averaging on Time-series Noise

                                                              Note the
                                                              dominance of
                                                              correlated errors
                                                              and unrealistic
                                                              rate uncertainties
                                                              with a white
                                                               .01 mm/yr N,E
                                                              .04 mm/yr U

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          Yellow: Daily (raw)   Blue: 7-day averages
Same site, East component   ( daily wrms 0.9 mm nrms 0.5 )

                                                             64-d avg
                                                             wrms 0.7 mm
                                                             nrms 2.0

                                                             100-d avg
                                                             wrms 0.6 mm
                                                             nrms 3.4

                                                             400-d avg
                                                             wrms 0.3 mm
                                                             nrms 3.1

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Estimating a “realistic-sigma” by fitting an exponential function to
chi-square vs averaging time

                                                         Get scale factor by
                                                         evaluating the function
                                                         at an infinite averaging

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Using TSVIEW to compute and display the “realistic-sigma” results

                                                                                   Note rate
                                                                                   with the
                                                                                   algorithm :

                                                                                   0.09 mm/yr N
                                                                                   0.13 mm/yr E
                                                                                   0.13 mm/yr U

Red lines show the 68% probability bounds of the velocity based on the results of applying the
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     Comparison of estimated velocity uncertainties using spectral
     analysis (CATS*) and Gauss-Markov fitting of averages (GLOBK)

  Plot courtesy E. Calias
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 Velocity Errors due to Seasonal Signals
          Annual signals from atmospheric and hydrological loading, monument translation and
          tilt, and antenna temperature sensitivity are common in GPS time series
                                      Theoretical analysis of a continuous time
                                        series by Blewitt and Lavallee [2002]

                                      Top: Bias in velocity from a 1mm
                                         sinusoidal signal in-phase and with a
                                         90-degree lag with respect to the start
                                         of the data span

                                      Bottom: Maximum and rms velocity bias
                                         over all phase angles
                                            – The minimum bias is NOT obtained with
                                              continuous data spanning an even
                                              number of years
                                            – The bias becomes small after 3.5 years
                                              of observation

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  Summary of Practical Approaches
• White noise + flicker noise (+ random walk) to model the
  spectrum [Williams et al., 2004]
• White noise as a proxy for flicker noise [Mao et al., 1999]
• Random walk to model to model an exponential spectrum
  [Herring “realistic sigma” algorithm for velocities]
• “Eyeball” white noise + random walk for non-continuous
• Only the last two can be applied in GLOBK for velocity
• All approaches require common sense and verification

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  Determining the Uncertainties of GPS
     Estimates of Station Velocities
• Understanding the sources of error
• Time series analysis to determine statistics for
  reweighting the data
• Whatever the assumed error model and tools
  used to implement it, external validation is

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     External Validation for Velocity Uncertainties
      -- assume no strain within a geological rigid block

                                                             GMT plot at
                                                             17 sites in
                                                             4-5 velocities
                                                             pierce error

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     .. same solution plotted with 95% confidence ellipses

                                                             1-2 of 17
                                                             pierce error

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    A more rigorous assessment for data from Cascadia

                                                    Colors show slipping
                                                    and locked portions
                                                    of the subducting
                                                    slab where the
                                                    surface velocities are
                                                    highly sensitive to
                                                    the model; area to
                                                    the east is slowly
                                                    deforming and
                                                    insensitive to the
                                                    details of the model

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                                                        McCaffrey et al. 2007
                                 Velocities and
                                 70% error
                                 ellipses for 300
                                 sites observed by
                                 continuous and
                                 GPS 1991-2004

                                 Test area (next
                                 slide) is east of

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                                 Residuals to elastic block
                                 model for 70 sites in
                                 slowly deforming region

                                 Error ellipses are for 70%
                                 13-17 velocities pierce
                                 their ellipse

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                                               Locations of one
                                               continuous (BURN)
          217U                                 and 3 survey-mode
                                               sites for time series
                                               shown in next

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                                 Time series of monthly
                                 position estimates for
                                 continuous site BURN

                                 Wrms ~ 1 mm N,E
                                     ~ 3 mm U

                                 Rate uncertainties
                                  < 0.2 mm/yr N,E
                                     0.7 mm/yr U
                                 do not include random
                                 walk added for velocity

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                                               Next slide shows
                                               time series for
          217U                                 survey-mode site

                                               Note consistency
          DALL                                 with nearby sites

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          Time series for survey-mode site 217U

                                                          estimates based
                                                          on 8-24 hr

                                                          Note < 1 mm
                                                          due to 7-yr time

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                                               Next slide shows
                                               time series for
          217U                                 survey-mode site


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          Horizontal time series for survey-mode site SARG

                                                       Position estimates
                                                       based on 8-24 hr

                                                       Note 1 mm wrms and
                                                             < 1 mm rate sigmas

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                                               Next slide shows
                                               time series for
          217U                                 survey-mode site

                                               Note consistency
          DALL                                 with nearby sites
                                               except continuous
                                               site GWEN

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          Horizontal time series for survey-mode site DALL

                                                       Position estimates
                                                       based on 8-24 hr

                                                       Rate sigmas < 1 mm/yr
                                                       and consistent with
                                                       surrounding sites even
                                                       with velocities
                                                       determined essentially
                                                       by two occupations 3
                                                       yrs apart

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          Statistics of Time Series

                                              Distribution of normalized rms
                                              for horizontal magnitudes
                                              residuals after removing the
                                              block model

                                              357 sites

                                               E, N 1.00, 1.03


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                       Statistics of Velocity Residuals

                                                                     Cumulative histogram of
                                                                     normalized velocity
                                                                     residuals for Eastern
                                                                     Oregon & Washington     (
          Percent                                                    70 sites )
                                                                     Noise added to position for
                                                                     each survey:
                                                                      0.5 mm random
                                                                     1.0 mm/sqrt(yr)) random walk

                                                                     Solid line is theoretical for
                                                                     Gaussian distribution

                    Ratio (velocity magnitude/uncertainty)

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                             Statistics of Velocity Residuals

          Within                                                  Same as last slide but with a
          Ratio                                                   smaller random-walk noise
                                                                  added :

                                                                  0.5 mm random
                                                                  0.5 mm/yr random walk

                                                                  ( vs 1.0 mm/sqrt(yr)) RW for
                                                                  ‘best’ noise model )

                                                                  Note greater number of
                                                                  residuals in range of 1.5-2.0

                    Ratio (velocity magnitude/uncertainty)

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                 Statistics of Velocity Residuals

                                                           Same as last slide but with larger
                                                           random and random-walk noise added
      Ratio                                                :

                                                            2.0 mm white noise
                                                            1.5 mm/sqrt(yr)) random walk

                                                           ( vs 0.5 mm WN and 1.0 mm/sqrt(yr))
                                                           RW for ‘best’ noise model )

                                                           Note smaller number of residuals in all
                                                           ranges above 0.1-sigma

                Ratio (velocity magnitude/uncertainty)

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• All algorithms for computing estimates of standard
  deviations have various problems: Fundamentally, rate
  standard deviations are dependent on low frequency
  part of noise spectrum which is poorly determined.
• Assumptions of stationarity are often not valid
• “Realistic sigma” algorithm is a covenient and reliable
  appraoch to getting velocity uncertainties in globk
• Velocity residuals from a model, together with their
  uncertainties, can be used to validate the error model

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    Tools for Error Analysis in GAMIT/GLOBK
• GAMIT: AUTCLN reweight = Y (default) uses phase rms from postfit edit
  to reweight data with constant + elevation-dependent terms
      – rename ( eq_file) _XPS or _XCL to remove outliers
      – sig_neu adds white noise by station and span; useful for handling outliers
      – mar_neu adds random-walk noise: principal method for controlling velocity
      – In the gdl files, can rescale variances of an entire h-file: useful when
        combining solutions from with different sampling rates or from different
        programs (Bernese, GIPSY)
• Utilities
      – Realistic sigma” algorithm implemented in tsview (MATLAB) and enfit/ensum;
        sh_gen_stats generates mar_neu commands for globk based on the noise
      – sh_plotvel (GMT) allows setting of confidence level of error ellipses
      – sh_tshist and sh_velhist can be used to generate histograms of time series and

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• There are no absolute methods that ensure the correct
  error model can be determined for a set of data
• We attempt to determine with 1-sigma values, that
  68% of values will be within this range due to noise;
  with 2-sigma, 95% of values (1-d) even when the
  probability distribution is not Gaussian.
• The most under certain aspect is determining the
  nature of the temporal and spatial correlations in the
  results. Generally, large amounts of data are needed
  for this and the assumption of stationarity.

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Spectral Analysis
Langbein and Johnson [J. Geophys. Res., 102, 591, 1997]
 Zhang et al. [J. Geophys. Res., 102, 18035, 1997]
Mao et al. [J. Geophys. Res., 104, 2797, 1999]
Dixon et al. [Tectonics , 19, 1, 2000] Herring [GPS Solutions, 7, 194, 2003]
Williams [J. Geodesy, 76, 483, 2003]
Williams et al. [J. Geophys. Res. 109, B03412, 2004]
Langbein [J. Geophys. Res., 113, B05405, 2008]
Williams, S. [GPS Solutions, 12, 147, 2008]

Effect of seasonal terms on velocity estimates
Blewitt and Lavaellee [J. Geophys. Res. 107, 2001JB000570, 2002]

Realistic Sigma Algorithm
Herring [GPS Solutions, 7, 194, 2003]
Reilinger et al. [J. Geophys. Res., 111, B5, 2006]

Validation in velocity fields
McClusky et al. [J. Geophys. Res. 105, 5695, 2000]
McClusky et al. [Geophys. Res. Lett., 28, 3369, 2000]
Davis et al. [J. Geophys. Res. Lett. 2003GL016961, 2003]
McCaffrey et al., [Geophys J. Int., 2007.03371, 2007]

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